Tricky Trigonometry

Geometry Level pending

$$x$$ is a real number. Let $$S$$ be the sum of all the possible distinct values of $$\sin x$$ that satisfy the following equation:

$7 - 2\cos^2x - 11\sin x = 0$

$$S$$ can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

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